Highly abundant number Cuisenaire rods 8

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a highly abundant number is a

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...

with the property that the sum of its divisors (including itself) is greater than the sum of the divisors of any smaller natural number. Highly abundant numbers and several similar classes of numbers were first introduced by , and early work on the subject was done by . Alaoglu and Erdős tabulated all highly abundant numbers up to 10⁴, and showed that the number of highly abundant numbers less than any ''N'' is at least proportional to log² ''N''.

Formal definition and examples

Formally, a natural number ''n'' is called highly abundant

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

for all natural numbers ''m'' < ''n'', :

\sigma(n) > \sigma(m)

where σ denotes the

sum-of-divisors function In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (including ...

. The first few highly abundant numbers are : 1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60, ... . For instance, 5 is not highly abundant because σ(5) = 5+1 = 6 is smaller than σ(4) = 4 + 2 + 1 = 7, while 8 is highly abundant because σ(8) = 8 + 4 + 2 + 1 = 15 is larger than all previous values of σ. The only odd highly abundant numbers are 1 and 3.

Relations with other sets of numbers

Although the first eight

factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \t ...

s are highly abundant, not all factorials are highly abundant. For example, :σ(9!) = σ(362880) = 1481040, but there is a smaller number with larger sum of divisors, :σ(360360) = 1572480, so 9! is not highly abundant. Alaoglu and Erdős noted that all

superabundant number In mathematics, a superabundant number (sometimes abbreviated as SA) is a certain kind of natural number. A natural number ''n'' is called superabundant precisely when, for all ''m'' < ''n'' :\frac 6/5. Superabundant numbers were defined by . ...

s are highly abundant, and asked whether there are infinitely many highly abundant numbers that are not superabundant. This question was answered affirmatively by . Despite the terminology, not all highly abundant numbers are

abundant number In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. The ...

s. In particular, none of the first seven highly abundant numbers (1, 2, 3, 4, 6, 8, and 10) is abundant. Along with 16, the ninth highly abundant number, these are the only highly abundant numbers that are not abundant. 7200 is the largest

powerful number A powerful number is a positive integer ''m'' such that for every prime number ''p'' dividing ''m'', ''p''2 also divides ''m''. Equivalently, a powerful number is the product of a square and a cube, that is, a number ''m'' of the form ''m'' = ''a ...

that is also highly abundant: all larger highly abundant numbers have a prime factor that divides them only once. Therefore, 7200 is also the largest highly abundant number with an odd sum of divisors., pp. 464–466.

Notes

References

* * * {{Classes of natural numbers Divisor function Integer sequences